The Florida State University College of Arts and Sciences Anova for Parameter Dependent Nonlinear Pdes and Numerical Methods for the Stochastic Stokes Equations

This dissertation includes the application of analysis-of-variance (ANOVA) expansions to analyze solutions of parameter dependent partial differential equations and the analysis and finite element approximations of the Stokes equations with stochastic forcing terms. In the first part of the dissertation, the impact of parameter dependent boundary conditions on the solutions of a class of nonlinear PDEs is considered. Based on the ANOVA expansions of functionals of the solutions, the effects of different parameter sampling methods on the accuracy of surrogate optimization approaches to PDE constrained optimization is considered. The effects of the smoothness of the functional and the nonlinearity in the PDE on the decay of the higher-order ANOVA terms are studied. The concept of effective dimensions is used to determine the accuracy of the truncated ANOVA expansions. Demonstrations are given to show that whenever truncated ANOVA expansions of functionals provide accurate approximations, optimizers found through a simple surrogate optimization strategy are also relatively accurate. The effects of several parameter sampling strategies on the accuracy of the surrogate optimization method are also considered; it is found that for this sparse sampling application, the Latin hypercube sampling method has advantages over other well-known sampling methods. Although most of the results are presented and discussed in the context of surrogate optimization problems, they also apply to other settings such as stochastic ensemble methods and reduced-order modeling for nonlinear PDEs. In the second part of the dissertation, we study the numerical analysis of the Stokes equations driven by a stochastic process. The random processes we use are white noise, colored noise and the homogeneous Gaussian process. When the process is white noise, we deal with the singularity of matrix Green’s functions in the form of mild solutions with